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Full text of "Treatise On Analysis Vol-Ii"

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13 EXTENSIONS OF HERMITiAN OPERATORS 433 4 = 0 for all n. It follows that [/' is the Cayley transform of a closed Her- mitian operator H' of defect (1,0), and consequently H" = -H'is a closed Hermitian operator of defect (0, 1) (15.13.6). It can be shown that there exist closed Hermitian operators of defect (m, n) for all choices of m and n in the set N u {+00} (Problem 7). In Chapter XXIII we shall see, by means of the theory of distributions, how to describe the closure of the Hermitian operator H = z'D defined in (15.13.2). We shall show that the defect of H** is (1, 1) if the interval I is bounded, (1, 0) if I is bounded above but not below, (0, 1) if I is bounded below but not above, and finally (0, 0) if /= R. Theorem (15.13.6) generalizes as follows. (15.13.10) For every complex number X such that S(X)>Q9 the subspace E+(l) (resp. E~(A)) of dom(/f*) consisting of the solutions of H* • x = Ix (resp. of H* • x = —/be) is isomorphic to E# (resp. E#). Since the subspace G~ + T(H) of T(H*) is closed in E x E (15.13.6), the restriction H1 of H* to domCF^) = E# + dom(/f) is a closed operator. Put A = \i + fv, where ^ and v are real and v > 0, and calculate (K^ — A/) * jc||2 for some x = y + z e domC/^), where y e dom(# ) and z e E# . We have and since H* • z = — iz, we have (H * y \ z) = (y \ H* • z) = i(y \ z). Now, (15.13.10.1) \\(H, - U) • *||2 = \\(H, - id) ' x\\2 + v2 W2 ^ v2 ||*||2 by the preceding calculation and the hypothesis v > 0. If we put F = Im^ - II), it follows that H^ - U is a bijection of domCHi) onto F, and the inverse mapping (Hv — II)""1 is a continuous linear mapping of the normed space F into E; since it is also a closed operator (because fli is closed), we conclude (15.12.2) that the subspace F of E is closed in E. We shall show that in fact F = E. We assert first that this is the case when A = i: for this we have to prove that if ((H^ - //) * x \ z) = 0 for all x e dom^), then z must be— en+1 for all n ^ 0,